On an open question in recovering Sturm-Liouville-type operators with delay
aa r X i v : . [ m a t h . SP ] S e p ON AN OPEN QUESTION IN RECOVERING STURM–LIOUVILLE-TYPEOPERATORS WITH DELAYNebojˇsa Djuri´c and Sergey Buterin Abstract.
In recent years, there appeared a considerable interest in the inverse spectral theory forfunctional-differential operators with constant delay. In particular, it is well known that specificationof the spectra of two operators ℓ j , j = 0 , , generated by one and the same functional-differentialexpression − y ′′ ( x ) + q ( x ) y ( x − a ) under the boundary conditions y (0) = y ( j ) ( π ) = 0 uniquelydetermines the complex-valued square-integrable potential q ( x ) vanishing on (0 , a ) as soon as a ∈ [ π/ , π ) . For many years, it has been a challenging open question whether this uniqueness resultwould remain true also when a ∈ (0 , π/ . Recently, a positive answer was obtained for the case a ∈ [2 π/ , π/ . In this paper, we give, however, a negative answer to this question for a ∈ [ π/ , π/
1. Introduction
One of the first results in the inverse spectral theory says that the spectra of two boundary valueproblems for one and the same Sturm–Liouville equation with one common boundary condition: − y ′′ ( x ) + q ( x ) y ( x ) = λy ( x ) , y (0) = y ( j ) ( π ) = 0 , j = 0 , , (1)uniquely determine the potential q ( x ) , see [1], where also local solvability and, actually, stability ofthis inverse problem were established in the class of real-valued q ( x ) ∈ L (0 , π ) . Later on, these resultswere refined and generalized to other classes of potentials and boundary conditions [2–7]. Moreover,there appeared methods, which gave global solution for the inverse Sturm–Liouville problem as wellas for inverse problems for other classes of differential operators (see, e.g., monographs [7–10]).In recent years, there was a considerable interest in inverse problems also for Sturm–Liouville-typeoperators with deviating argument (see [11–29] and references therein), which are often more adequatefor modelling various real world processes frequently possessing a nonlocal nature.For j = 0 , , denote by { λ n,j } the spectrum of the boundary value problem L j ( a, q ) of the form − y ′′ ( x ) + q ( x ) y ( x − a ) = λy ( x ) , < x < π, (2) y (0) = y ( j ) ( π ) = 0 , (3)where q ( x ) is a complex-valued function in L (0 , π ) and q ( x ) = 0 on (0 , a ) , while a ∈ (0 , π ) . Forfunctional-differential operators as well as for other classes of nonlocal ones, classical method of theinverse spectral theory for differential operators do not work. Consider the following inverse problem.
Inverse Problem 1.
Given the spectra { λ n, } and { λ n, } , find the potential q ( x ) . Instead of (3), one could alternatively impose the boundary conditions y ′ (0) − hy (0) = y ( j ) ( π ) = 0 (4)with a complex parameter h, which can be found given the input data of Inverse Problem 1. Variousaspects of Inverse Problem 1 were studied in [11, 12, 14–22, 24, 26, 27] and other works. Faculty of Architecture, Civil Engineering and Geodesy, University of Banja Luka, [email protected] Department of Mathematics, Saratov State University, [email protected] he values π/ π/ a are critical for the solution of this inverseproblem. Specifically, in the case a ≥ π/ , the dependence of the characteristic function of eachproblem L j ( a, q ) on the potential q ( x ) is linear, while for a < π/ { n k } k ≥ were obtained that are necessary and sufficient for the unique determination of the potential q ( x ) by specifying the corresponding subspectra { λ n k , } and { λ n k , } . The nonlinear case a < π/ a ∈ (0 , π ) , it was established that if the given spectra coincide with the spectra of the correspondingproblems with the zero potential, then q ( x ) is zero too.Under the general settings, i.e. for arbitrary nonzero potentials, it was a long term challenging openquestion whether the uniqueness of solution of Inverse Problem 1 takes place for a < π/ . Recently,a positive answer to this question was given independently in [18] (for the conditions (3)) and in [20](for (4)) as soon as a ∈ [2 π/ , π/ . In the present paper, we give a negative answer to the open question formulated above in thecase when a ∈ [ π/ , π/ . Namely, for each such a, we construct an infinite family of differentiso-bispectral potentials q ( x ) , i.e. of those for which the problems L ( a, q ) and L ( a, q ) have oneand the same pair of spectra. This looks quite unusual in light of the classical results for the caseof a = 0 . Our counterexample appears even more unexpected under consideration that the recentpaper [27] announces that specification of the spectra of both boundary value problems consistingof (2) and (4) uniquely determines the potential q ( x ) also for a ∈ [ π/ , π/ . Although in [27]Robin boundary conditions were imposed also in the point π, they can be easily reduced to (4). InSection 3, we discuss why our result does not refute the uniqueness theorem in [27] for a ∈ [ π/ , π/
2. The main result
Fix a ∈ [ π/ , π/ . For a nonzero real-valued function h ( x ) ∈ L (5 a/ , π ) , we consider theintegral operator M h f ( x ) = Z π − x + a a K h (cid:16) x + t − a (cid:17) f ( t ) dt, a < x < π − a, where K h ( x ) = Z πx h ( τ ) dτ. (5)Thus, M h is a nonzero compact Hermitian operator acting in the space L (3 a/ , π − a ) . Hence, ithas at least one nonzero real eigenvalue η. We assume that η = 1 , which can always be achieved bychoosing h ( x ) /η instead of h ( x ) . Let e ( x ) be the corresponding eigenfunction, i.e. M h e ( x ) = e ( x ) , a < x < π − a. (6)Consider the one-parametric family of potentials B := { q α ( x ) } α ∈ C determined by the formula q α ( x ) = , x ∈ (cid:16) , a (cid:17) ∪ ( π − a, a ) ∪ (cid:16) π − a , a (cid:17) ,αe ( x ) , x ∈ (cid:16) a , π − a (cid:17) , − αK h (cid:16) x + a (cid:17) Z x − a a e ( t ) dt, x ∈ (cid:16) a, π − a (cid:17) ,h ( x ) , x ∈ (cid:16) a , π (cid:17) . (7)The main result of the present paper is the following theorem.2 heorem 1. For j = 0 , , the spectrum { λ n,j } of the boundary value problem L j ( a, q α ) doesnot depend on α. Theorem 1 means that the problems L ( a, q α ) and L ( a, q α ) have one and the same pair ofspectra { λ n, } and { λ n, } for all values of the parameter α ∈ C , i.e. the solution of InverseProblem 1 is not unique. We note that, since the zero potential does not belong to B, Theorem 1does not contradict the uniqueness result in [12]. The proof of Theorem 1 is given in the next section.
Remark 1.
This reminds the situation of the following operator with frozen argument: ℓy = − y ′′ ( x ) + q ( x ) y ( b ) , y ( α ) (0) = y ( β ) (1) = 0 , α, β ∈ { , } , b ∈ [0 , . It is known that the unique recoverability of q ( x ) from the spectrum of ℓ depends on the triple ofparameters ( α, β, b ) . In [28], one can find a complete description of all degenerate and non-degeneratecases for rational values of b, while for irrational ones the uniqueness always takes place, see [29].Theorem 1 opens the analogous type of questions for Inverse Problem 1, which consist in givingdescription of ranges of the delay parameter a along with the types of boundary conditions, forwhich the uniqueness of solution of Inverse Problem 1 takes place. In the opposite degenerate cases,it would be interesting to describe classes of iso-bispectral potentials. It is especially important toinvestigate the case of arbitrarily small values of a making the problem ”close” to the classical case (1).
3. Proof of Theorem 1
Let S ( x, λ ) be a solution of equation (2) under the initial conditions S (0 , λ ) = 0 and S ′ (0 , λ ) = 1 . For j = 0 , , eigenvalues of the problem L j ( a, q ) coincide with zeros of its characteristic function∆ j ( λ ) = S ( j ) ( π, λ ) . Thus, the spectrum of any problem L j ( a, q ) does not depend on q ( x ) ∈ B forsome subset B ⊂ L (0 , π ) as soon as neither does the corresponding characteristic function ∆ j ( λ ) . Put ρ = λ and denote ω := Z πa q ( x ) dx. (8)The following representations hold (see, e.g., [26]):∆ ( λ ) = sin ρπρ − ω cos ρ ( π − a )2 ρ + 12 ρ Z πa w ( x ) cos ρ ( π − x + a ) dx, (9)∆ ( λ ) = cos ρπ + ω sin ρ ( π − a )2 ρ − ρ Z πa w ( x ) sin ρ ( π − x + a ) dx, (10)where the function w ( x ) is determined by the following formula for k = 0 : w k ( x ) = q ( x ) , x ∈ (cid:16) a, a (cid:17) ∪ (cid:16) π − a , π (cid:17) ,q ( x ) + Q k ( x ) , x ∈ (cid:16) a , π − a (cid:17) , (11)while Q k ( x ) = Z π − ax − a (cid:16) q ( t + a ) Z ta q ( τ ) dτ − q ( t ) Z πt + a q ( τ ) dτ − ( − k Z πt + a q ( τ − t ) q ( τ ) dτ (cid:17) dt. (12)Note that, since the function ∆ ( λ ) is entire in λ, representation (9) implies ω = Z πa w ( x ) dx, (13)which can also be checked independently by direct calculation using (11) and (12) for k = 0 . Thus,the spectrum of L j ( a, q ) , j = 0 , , is independent of q ( x ) ∈ B if so is the function w ( x ) .
3y changing the order of integration in (12), it is easy to obtain the representation Q k ( x ) = Z x − a a q ( t ) dt Z πx + a q ( τ ) dτ − ( − k Z π − x + a a q ( t ) dt Z πx + t − a q ( τ ) dτ, (14)where x ∈ (3 a/ , π − a/ . Let q ( x ) = 0 on ( a, a/ . Then (14) takes the form Q k ( x ) = − ( − k M q q ( x ) , x ∈ (cid:16) a , π − a (cid:17) , , x ∈ ( π − a, a ) ,K q (cid:16) x + a (cid:17) Z x − a a q ( t ) dt, x ∈ (cid:16) a, π − a (cid:17) , (15)where q ( x ) = q ( x ) , x ∈ (cid:16) a , π − a (cid:17) , q ( x ) = q ( x ) , x ∈ (cid:16) a , π (cid:17) , while M h and K h ( x ) are determined by (5). Thus formulae (11) and (15) give w k ( x ) = , x ∈ (cid:16) a, a (cid:17) ,q ( x ) − ( − k M q q ( x ) , x ∈ (cid:16) a , π − a (cid:17) ,q ( x ) , x ∈ ( π − a, a ) ,q ( x ) + K q (cid:16) x + a (cid:17) Z x − a a q ( t ) dt, x ∈ (cid:16) a, π − a (cid:17) ,q ( x ) , x ∈ (cid:16) π − a , a (cid:17) ,q ( x ) , x ∈ (cid:16) a , π (cid:17) . (16)Substituting q ( x ) = q α ( x ) into (16) for k = 0 , where q α ( x ) is determined by (7), and taking (6)into account, we arrive at w ( x ) = , x ∈ (cid:16) a, a (cid:17) ,h ( x ) , x ∈ (cid:16) a , π (cid:17) . Thus, according to (9), (10) and (13), the characteristic function ∆ j ( λ ) of the problem L j ( a, q α ) foreach j = 0 , α, which finishes the proof of Theorem 1. (cid:3) Remark 2.
Thus, we have constructed a class B of potentials on those the function w ( x ) ap-pearing in representations (9) and (10) does not depend. By virtue of relation (13), this independenceis inherited by both characteristic functions ∆ ( λ ) and ∆ ( λ ) , which gives Theorem 1.Let us show why this strategy fails (at least, in the present form) in the case of boundary con-ditions (4). For j = 0 , , denote by M j ( a, h, q ) the boundary value problem for equation (2) withboundary conditions (4). Then eigenvalues of the problem M j ( a, , q ) coincide with zeros of itscharacteristic function Θ j ( λ ) := C ( j ) ( π, λ ) , where C ( x, λ ) is the solution of (2) under the initialconditions C (0 , λ ) = 1 and C ′ (0 , λ ) = 0 . Analogously to (9) and (10), one can obtain the represen-tations (see, e.g., [20]):Θ ( λ ) = cos ρπ + ω sin ρ ( π − a )2 ρ + 12 ρ Z πa w ( x ) sin ρ ( π − x + a ) dx, (17)Θ ( λ ) = − ρ sin ρπ + ω ρ ( π − a ) + 12 Z πa w ( x ) cos ρ ( π − x + a ) dx, (18)4here ω is determined by formula (8), while the function w ( x ) has the form (11) for k = 1 . Anal-ogously to B, one can construct a family B of potentials p α ( x ) on those the function w ( x ) doesnot depend. Indeed, for this purpose, the same scheme can be used but involving the operator − M h instead of M h . However, the main difference from the case of boundary conditions (3) is that, in thecase of (4), there is no relation analogous to (13). In other words, the constant ω is not determinedby the function w ( x ) . Thus, each characteristic function (17) and (18) could depend on α even if w ( x ) did not. So the presented scheme does not refute the uniqueness theorem in [27]. Acknowledgement.
The first author was supported by the Project 19.032/961-103/19 of theRepublic of Srpska Ministry for Scientific and Technological Development, Higher Education andInformation Society. The second author was supported by Grants 19-01-00102 and 20-31-70005 of theRussian Foundation for Basic Research.
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